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Theorem nbuhgr 26126
Description: The set of neighbors of a vertex in a hypergraph. This version of nbgrval 26121 (with 𝑁 being an arbitrary set instead of being a vertex) only holds for classes whose edges are subsets of the set of vertices (hypergraphs!). (Contributed by AV, 26-Oct-2020.) (Proof shortened by AV, 15-Nov-2020.)
Hypotheses
Ref Expression
nbgrel.v 𝑉 = (Vtx‘𝐺)
nbgrel.e 𝐸 = (Edg‘𝐺)
Assertion
Ref Expression
nbuhgr ((𝐺 ∈ UHGraph ∧ 𝑁𝑋) → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒})
Distinct variable groups:   𝑒,𝑛   𝑒,𝐸   𝑒,𝐺   𝑒,𝑁   𝑒,𝑉   𝑛,𝐺   𝑛,𝑁   𝑛,𝑉   𝑒,𝑋,𝑛
Allowed substitution hint:   𝐸(𝑛)

Proof of Theorem nbuhgr
StepHypRef Expression
1 nbgrel.v . . . 4 𝑉 = (Vtx‘𝐺)
2 nbgrel.e . . . 4 𝐸 = (Edg‘𝐺)
31, 2nbgrval 26121 . . 3 (𝑁𝑉 → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒})
43a1d 25 . 2 (𝑁𝑉 → ((𝐺 ∈ UHGraph ∧ 𝑁𝑋) → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒}))
5 df-nel 2894 . . . . . 6 (𝑁𝑉 ↔ ¬ 𝑁𝑉)
61nbgrnvtx0 26124 . . . . . 6 (𝑁𝑉 → (𝐺 NeighbVtx 𝑁) = ∅)
75, 6sylbir 225 . . . . 5 𝑁𝑉 → (𝐺 NeighbVtx 𝑁) = ∅)
87adantr 481 . . . 4 ((¬ 𝑁𝑉 ∧ (𝐺 ∈ UHGraph ∧ 𝑁𝑋)) → (𝐺 NeighbVtx 𝑁) = ∅)
9 simpl 473 . . . . . . . . . . . 12 ((𝐺 ∈ UHGraph ∧ 𝑁𝑋) → 𝐺 ∈ UHGraph )
109adantr 481 . . . . . . . . . . 11 (((𝐺 ∈ UHGraph ∧ 𝑁𝑋) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) → 𝐺 ∈ UHGraph )
112eleq2i 2690 . . . . . . . . . . . 12 (𝑒𝐸𝑒 ∈ (Edg‘𝐺))
1211biimpi 206 . . . . . . . . . . 11 (𝑒𝐸𝑒 ∈ (Edg‘𝐺))
13 edguhgr 25919 . . . . . . . . . . 11 ((𝐺 ∈ UHGraph ∧ 𝑒 ∈ (Edg‘𝐺)) → 𝑒 ∈ 𝒫 (Vtx‘𝐺))
1410, 12, 13syl2an 494 . . . . . . . . . 10 ((((𝐺 ∈ UHGraph ∧ 𝑁𝑋) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) ∧ 𝑒𝐸) → 𝑒 ∈ 𝒫 (Vtx‘𝐺))
15 selpw 4137 . . . . . . . . . . . 12 (𝑒 ∈ 𝒫 (Vtx‘𝐺) ↔ 𝑒 ⊆ (Vtx‘𝐺))
161eqcomi 2630 . . . . . . . . . . . . 13 (Vtx‘𝐺) = 𝑉
1716sseq2i 3609 . . . . . . . . . . . 12 (𝑒 ⊆ (Vtx‘𝐺) ↔ 𝑒𝑉)
1815, 17bitri 264 . . . . . . . . . . 11 (𝑒 ∈ 𝒫 (Vtx‘𝐺) ↔ 𝑒𝑉)
19 sstr 3591 . . . . . . . . . . . . . . 15 (({𝑁, 𝑛} ⊆ 𝑒𝑒𝑉) → {𝑁, 𝑛} ⊆ 𝑉)
20 vex 3189 . . . . . . . . . . . . . . . . 17 𝑛 ∈ V
21 prssg 4318 . . . . . . . . . . . . . . . . . 18 ((𝑁𝑋𝑛 ∈ V) → ((𝑁𝑉𝑛𝑉) ↔ {𝑁, 𝑛} ⊆ 𝑉))
2221bicomd 213 . . . . . . . . . . . . . . . . 17 ((𝑁𝑋𝑛 ∈ V) → ({𝑁, 𝑛} ⊆ 𝑉 ↔ (𝑁𝑉𝑛𝑉)))
2320, 22mpan2 706 . . . . . . . . . . . . . . . 16 (𝑁𝑋 → ({𝑁, 𝑛} ⊆ 𝑉 ↔ (𝑁𝑉𝑛𝑉)))
24 simpl 473 . . . . . . . . . . . . . . . 16 ((𝑁𝑉𝑛𝑉) → 𝑁𝑉)
2523, 24syl6bi 243 . . . . . . . . . . . . . . 15 (𝑁𝑋 → ({𝑁, 𝑛} ⊆ 𝑉𝑁𝑉))
2619, 25syl5com 31 . . . . . . . . . . . . . 14 (({𝑁, 𝑛} ⊆ 𝑒𝑒𝑉) → (𝑁𝑋𝑁𝑉))
2726ex 450 . . . . . . . . . . . . 13 ({𝑁, 𝑛} ⊆ 𝑒 → (𝑒𝑉 → (𝑁𝑋𝑁𝑉)))
2827com13 88 . . . . . . . . . . . 12 (𝑁𝑋 → (𝑒𝑉 → ({𝑁, 𝑛} ⊆ 𝑒𝑁𝑉)))
2928ad3antlr 766 . . . . . . . . . . 11 ((((𝐺 ∈ UHGraph ∧ 𝑁𝑋) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) ∧ 𝑒𝐸) → (𝑒𝑉 → ({𝑁, 𝑛} ⊆ 𝑒𝑁𝑉)))
3018, 29syl5bi 232 . . . . . . . . . 10 ((((𝐺 ∈ UHGraph ∧ 𝑁𝑋) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) ∧ 𝑒𝐸) → (𝑒 ∈ 𝒫 (Vtx‘𝐺) → ({𝑁, 𝑛} ⊆ 𝑒𝑁𝑉)))
3114, 30mpd 15 . . . . . . . . 9 ((((𝐺 ∈ UHGraph ∧ 𝑁𝑋) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) ∧ 𝑒𝐸) → ({𝑁, 𝑛} ⊆ 𝑒𝑁𝑉))
3231rexlimdva 3024 . . . . . . . 8 (((𝐺 ∈ UHGraph ∧ 𝑁𝑋) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) → (∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒𝑁𝑉))
3332con3rr3 151 . . . . . . 7 𝑁𝑉 → (((𝐺 ∈ UHGraph ∧ 𝑁𝑋) ∧ 𝑛 ∈ (𝑉 ∖ {𝑁})) → ¬ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒))
3433expdimp 453 . . . . . 6 ((¬ 𝑁𝑉 ∧ (𝐺 ∈ UHGraph ∧ 𝑁𝑋)) → (𝑛 ∈ (𝑉 ∖ {𝑁}) → ¬ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒))
3534ralrimiv 2959 . . . . 5 ((¬ 𝑁𝑉 ∧ (𝐺 ∈ UHGraph ∧ 𝑁𝑋)) → ∀𝑛 ∈ (𝑉 ∖ {𝑁}) ¬ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒)
36 rabeq0 3931 . . . . 5 ({𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒} = ∅ ↔ ∀𝑛 ∈ (𝑉 ∖ {𝑁}) ¬ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒)
3735, 36sylibr 224 . . . 4 ((¬ 𝑁𝑉 ∧ (𝐺 ∈ UHGraph ∧ 𝑁𝑋)) → {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒} = ∅)
388, 37eqtr4d 2658 . . 3 ((¬ 𝑁𝑉 ∧ (𝐺 ∈ UHGraph ∧ 𝑁𝑋)) → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒})
3938ex 450 . 2 𝑁𝑉 → ((𝐺 ∈ UHGraph ∧ 𝑁𝑋) → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒}))
404, 39pm2.61i 176 1 ((𝐺 ∈ UHGraph ∧ 𝑁𝑋) → (𝐺 NeighbVtx 𝑁) = {𝑛 ∈ (𝑉 ∖ {𝑁}) ∣ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wnel 2893  wral 2907  wrex 2908  {crab 2911  Vcvv 3186  cdif 3552  wss 3555  c0 3891  𝒫 cpw 4130  {csn 4148  {cpr 4150  cfv 5847  (class class class)co 6604  Vtxcvtx 25774  Edgcedg 25839   UHGraph cuhgr 25847   NeighbVtx cnbgr 26111
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-1st 7113  df-2nd 7114  df-edg 25840  df-uhgr 25849  df-nbgr 26115
This theorem is referenced by:  uhgrnbgr0nb  26137
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